Domain and Range of Relations
Domain and Range of Relations: Let R be a relation from A to B. The domain of R is the set of all those elements a ϵ A such that (a, b) ϵ R for some b ϵ R for some b ϵ B.
Domain of R = {a ϵ A: (a, b) ϵ R, ∀ b ϵ B}
And range of R is the set of all those elements b ϵ B such that (a, b) ϵ R for some b ϵ B such that (a, b) ϵ R for some a ϵ A.
Range of R = {b ϵ B: (a, b) ϵ R, ∀ a ϵ R},
Here, B is called the codomain of R
Example: Let A = {1, 2, 3} & B = {3, 5, 6}
aRb ⇒ a< b then, R = {(1, 5), (2, 5), (3, 5), (1, 6), (2, 6), (3, 6)}
Domain of R = {1, 2, 3}
Rage of R = {5, 6}
And codomain of R = {3, 5, 6}
Note: Let A and B be two non empty finite sets having p and q elements respectively
Total number of relations from A to B = 2pq
Example 1: find the domain and range of the relation R defined by R = {x, x + 5}: x ϵ {0, 1, 2, 3, 4, 5}.
Solution: Given that R = {x, x+5}: x ϵ {0, 1, 2, 3, 4, 5} … (1)
Putting x = 0, 1, 2, 3, 4, 5, we get
Domain = {0, 1, 2, 3, 4, 5}
After putting the value of x from equation (1), we get
y = x + 5
= 5, 6, 7, 8, 9, 10
Range = {5, 6, 7, 8, 9, 10}
Example 2: The relation R defined on the best of natural numbers as {(a, b): a differs from b by 3}, is given by
Solution: Given that
{(a, b): a differs from b by 3}
{(a, b): a – b = 3}
Put a = 4, 5, 6, . . .
a = 4
4 – b = 3
b = 1
(a, b) = (4, 1)
a = 5
5 – b = 3
b = 2
(a, b) = (5, 2)
a = 6
6 – b = 3
b = 3
(a, b) = (6, 3)
= {(4, 1), (5, 2), (6, 3), . . .}.